, Volume 62, Issue 3, pp 311325
First online:
The linear arboricity of graphs
 N. AlonAffiliated withDepartment of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University
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Alinear forest is a forest in which each connected component is a path. Thelinear arboricity la(G) of a graphG is the minimum number of linear forests whose union is the set of all edges ofG. Thelinear arboricity conjecture asserts that for every simple graphG with maximum degree Δ=Δ(G),\(la(G) \leqq [\frac{{\Delta + 1}}{2}].\). Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(G) is la(G)≦⌈3Δ/5⌉ for even Δ and la(G)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for everyɛ>0 there is a Δ_{0}=Δ_{0}(ɛ) so that la(G)≦(1/2+ɛ)Δ for everyG with maximum degree Δ≧Δ_{0}. To do this, we first prove the conjecture for everyG with an even maximum degree Δ and withgirth g≧50Δ.
 Title
 The linear arboricity of graphs
 Journal

Israel Journal of Mathematics
Volume 62, Issue 3 , pp 311325
 Cover Date
 198810
 DOI
 10.1007/BF02783300
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 N. Alon ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel